About

The half-crossover method is a technique for quickly calculating the pip count over the board.

The main benefits are that (1) it requires simpler mental calculations, resulting in fewer errors for most people, and (2) it quickly provides an approximate pip count (median absolute error 3 pips), which is good enough for many backgammon decisions -- allowing you to only spend the few more seconds needed to get the exact pip count in positions where it's necessary.

The main downside of the half-crossover method is that it works best for positions with all 15 checkers on the board. There is an adjustment for when checkers start getting borne off, but generally in these positions it is easier to just use cluster counting.

The half-crossover method was first described by Doug Zare in 2000 (original article).

The method

The board is divided into eight half-crossover zones of three points each, numbered −1 to 6. Each checker contributes the specified number of half-crossovers to the count:

To count the pips for one side:

Checkers on the bar

Checkers on the bar count as 7 half-crossovers. It's rare to need an exact count in a position where there are checkers on the bar, but if you do, the bar acts as a light point (i.e., it subtracts 1 from the pip count in Step 3), since it is effectively the 25-point.

Borne-off checkers

In positions where checkers have been borne off, the remaining checkers are generally concentrated in the inner board, where cluster counting tend to be most effective. But the half-crossover method can be used: just count as usual (with no contribution from checkers in the bearoff tray), then subtract 5 from the final count for every borne-off checker.

Tips

1. Approximate vs. exact counts

In most positions you don't need an exact count - just who is ahead and by roughly how much. A major advantage of the half-crossover method is that you can count half-crossovers for each side (i.e., Step 1, which takes just a few seconds with practice) then multiply the difference by 3 pips per half-crossover to get an approximate pip count difference. The median absolute error of this approach is only 3 pips, so if you don't need a precise pip count, this can save a lot of time.

2. 3X + 75

It's worth practicing quickly and accurately calculating 3X + 75 for X in the range of approximately −5 to 20. Drill 2 allows you to practice this in isolation.

For large X (>10), you may find it easier to calculate 3X + 75 as (3X + 100) − 25. For example, X = 14 → 3X = 42 → 42 + 100 = 142 → 142 − 25 = 117. But eventually (with practice) you can just mentally say "14, 42, 142, 117", then eventually (with more practice) instantly map 14 half crossovers  → 117 pips.

3. Patterns

Eventually you'll start to recognize common patterns of checkers that you can immediately recognize as a certain number of half-crossovers. No need to memorize (these will come naturally with time), but here are some to look out for:

14 half-crossovers

The common formation of two checkers each on the 18-point and midpoint will eventually be recognizable as an instant 14 half-crossovers:

20 half-crossovers

Similarly, you will eventually recognize two checkers on the 18-point and four on the midpoint as an instant 20 half-crossovers:

Mirrors: 5 half-crossovers per pair

Two checkers in opposing zones (across the board) always sum to 5 half-crossovers.

Diagonals: 4 half-crossovers per pair

Two checkers in diagonal zones (with the far-side checker farther from the bearoff tray side) always sum to 4 half-crossovers.

Reverse diagonals: 6 half-crossovers per pair

Two checkers in diagonal zones (with the far-side checker closer to the bearoff tray side) always sum to 6 half-crossovers.